A071989 a(n) = n-th decimal digit of the fractional part of the square root of the n-th nonsquare number (A000037).
4, 3, 6, 4, 5, 7, 6, 9, 5, 4, 7, 7, 6, 8, 3, 3, 0, 4, 5, 9, 8, 3, 3, 0, 8, 9, 6, 2, 4, 1, 0, 4, 4, 0, 6, 7, 9, 5, 1, 7, 4, 1, 3, 5, 7, 5, 7, 7, 4, 8, 8, 9, 5, 0, 5, 0, 6, 5, 1, 7, 3, 3, 9, 9, 7, 7, 6, 1, 4, 9, 9, 2, 7, 8, 5, 8, 4, 9, 4, 5, 4, 2, 8, 0, 2, 1, 7, 7, 4, 7, 4, 8, 1, 8, 4, 5, 7, 5, 8, 0, 0, 0, 1, 4, 3
Offset: 1
Examples
Sqrt(2)=1.4142135... -> the 1st decimal digit is 4, sqrt(3)=1.7320508... -> the 2nd decimal digit is 3, sqrt(5)=2.2360679... -> the 3rd decimal digit is 6, sqrt(6)=2.4494897... -> the 4th decimal digit is 4, etc.
References
- Martin Aigner & Günter M. Ziegler, Proofs from THE BOOK, Second Edition, Springer-Verlag, Berlin Heidelberg NY, Section of Analysis, Chptr 15, "Sets, function, and the continuum hypothesis", 2000, pp. 87-98.
- Georg Cantor, Über eine Eigenschaft des Inbegriffes aller reellen Zahlen ("On the Characteristic Property of All Real Numbers").
- Timothy Gowers, Editor, with June Barrow-Green & Imre Leader, Assc. Editors, The Princeton Companion to Mathematics, Princeton Un. Press, Princeton & Oxford, 2008, pp. 171 & 779.
- Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §7.5 Transfinite Numbers, pp. 257-262.
Links
- Chai Wah Wu, Table of n, a(n) for n = 1..10000
- Richard Lipton, Gödel's Lost Letter and P=NP.
- Luke Mastin, 19th Century Mathematics - Cantor.
- Tom Schaffter, Cantor's Diagonal Argument: Proof and Paradox.
- Eric Weisstein's World of Mathematics, Cantor Diagonal Method.
- Wikipedia, Cantor's diagonal argument.
Programs
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Mathematica
q[n_] := (m = Floor[n + Sqrt[n + Sqrt[n]]]; Floor[ Mod[ 10^n*Sqrt[m], 10]]); Table[ q[n], {n, 1, 105}] -
Python
from math import isqrt def A071989(n): return isqrt(10**(n<<1)*(n+(k:=isqrt(n))+int(n>=k*(k+1)+1)))%10 # Chai Wah Wu, Jul 20 2024
Formula
a(n) = floor(sqrt(A000037(n))*10^n) mod 10. - Jason Yuen, Aug 20 2024
Comments